Let be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke’s generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov function is equivalent to the existence of a classical control-Lyapunov function. This property leads to a generalization of a result on the systems with integrator.
Keywords: asymptotic stabilizability, converse Lyapunov theorem, nonsmooth analysis, differential inclusion, Filippov and krasovskii solutions, feedback
@article{COCV_2001__6__593_0, author = {Rifford, Ludovic}, title = {On the existence of nonsmooth {control-Lyapunov} functions in the sense of generalized gradients}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {593--611}, publisher = {EDP-Sciences}, volume = {6}, year = {2001}, mrnumber = {1872388}, zbl = {1002.93058}, language = {en}, url = {http://www.numdam.org/item/COCV_2001__6__593_0/} }
TY - JOUR AU - Rifford, Ludovic TI - On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2001 SP - 593 EP - 611 VL - 6 PB - EDP-Sciences UR - http://www.numdam.org/item/COCV_2001__6__593_0/ LA - en ID - COCV_2001__6__593_0 ER -
%0 Journal Article %A Rifford, Ludovic %T On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2001 %P 593-611 %V 6 %I EDP-Sciences %U http://www.numdam.org/item/COCV_2001__6__593_0/ %G en %F COCV_2001__6__593_0
Rifford, Ludovic. On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 593-611. http://www.numdam.org/item/COCV_2001__6__593_0/
[1] Stabilization with relaxed controls. Nonlinear Anal. 7 (1983) 1163-1173. | MR | Zbl
,[2] Viability theory. Birkhäuser Boston Inc., Boston, MA (1991). | MR | Zbl
,[3] Differential Inclusions. Springer-Verlag (1984). | MR | Zbl
and ,[4] Set-valued analysis. Birkhäuser (1990). | MR | Zbl
and ,[5] New results and examples in nonlinear feedback stabilization. Systems Control Lett. 12 (1989) 437-442. | MR | Zbl
and ,[6] Feedback stabilization and Lyapunov functions. SIAM J. Control Optim. 39 (2000) 25-48. | MR | Zbl
, , and ,[7] Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). Republished as Classics Appl. Math. 5 (1990). | MR | Zbl
,[8] Asymptotic stability and smooth Lyapunov functions. J. Differential Equations 149 (1998) 69-114. | MR | Zbl
, and ,[9] Nonsmooth Analysis and Control Theory. Springer-Verlag, New York, Grad. Texts in Math. 178 (1998). | MR | Zbl
, , and ,[10] On the stabilization of some nonlinear control systems: Results, tools, and applications, in Nonlinear analysis, differential equations and control (Montreal, QC, 1998). Kluwer Acad. Publ., Dordrecht (1999) 307-367. | MR | Zbl
,[11] Some open problems in control theory, in Differential geometry and control (Boulder, CO, 1997). Providence, RI, Amer. Math. Soc. (1999) 149-162. | MR | Zbl
,[12] A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Systems Estim. Control 4 (1994) 67-84. | MR | Zbl
and ,[13] Multivalued Differential Equations. de Gruyter, Berlin (1992). | MR | Zbl
,[14] Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers (1988). | MR | Zbl
,[15] Robust Nonlinear Control Design. State-Space and Lyapunov Techniques. Birkhäuser (1996). | MR | Zbl
and ,[16] Backstepping design with nonsmooth nonlinearities, in Proc. of the IFAC Nonlinear Control Systems design symposium. Tahoe City, California (1995).
and ,[17] Discontinuous differential equations. I, II. J. Differential Equations 32 (1979) 149-170, 171-185. | MR | Zbl
,[18] Les fonctions d'appui de la jacobienne généralisée de Clarke et de son enveloppe plénière. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1275-1278. | Zbl
and ,[19] Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay. Stanford University Press, Stanford, California (1963). Translated by J.L. Brenner. | Zbl
,[20] On the inversion of Lyapunov's second theorem on stability of motion. Amer. Math. Soc. Transl. Ser. 2 24 (1956) 19-77. | Zbl
,[21] A Lyapunov characterization of robust stabilization. Nonlinear Anal. 37 (1999) 813-840. | MR | Zbl
and ,[22] A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim. 34 (1996) 124-160. | MR | Zbl
, and ,[23] Contributions to stability theory. Ann. of Math. (2) 64 (1956) 182-206. | MR | Zbl
,[24] Continuous selections. I. Ann. of Math. (2) 63 (1956) 361-382. | MR | Zbl
,[25] On assigning the derivative of a disturbance attenuation clf, in Proc. of the 37th IEEE conference on decision and control. Tampa, Florida (1998).
and ,[26] Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Control Optim. 39 (2000) 1043-1064. | MR | Zbl
,[27] Étude de quelques problèmes de stabilisation, Ph.D. Thesis. ENS de Cachan (1993).
,[28] A “universal” construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett. 13 (1989) 117-123. | Zbl
,[29] Mathematical Control Theory. Springer-Verlag, New York, Texts Appl. Math. 6 (1990) (Second Edition, 1998). | MR | Zbl
,[30] Stability and stabilization: Discontinuities and the effect of disturbances, in Nonlinear analysis, differential equations and control (Montreal, QC, 1998). Kluwer Acad. Publ., Dordrecht (1999) 551-598. | MR | Zbl
,[31] A smooth Lyapunov function from a class- estimate involving two positive semidefinite functions. ESAIM: COCV 5 (2000) 313-367. | Numdam | MR | Zbl
and ,[32] A Lyapunov description of stability in control systems. Nonlinear Anal. 13 (1989) 3-74. | MR | Zbl
,[33] Sufficient Lyapunov-like conditions for stabilization. Math. Control Signals Systems 2 (1989) 343-357. | MR | Zbl
,[34] A local stabilization theorem for interconnected systems. Systems Control Lett. 18 (1992) 429-434. | MR | Zbl
,[35] An extension of Artstein's theorem on stabilization by using ordinary feedback integrators. Systems Control Lett. 20 (1993) 141-148. | Zbl
,